Adaptive equalizer with decision directed constant modulus algorithm

ABSTRACT

An adaptive equalizer for use in blind equalization systems to compensate for transmission channel distortion and noise in a digital communication system quantizes the input signal samples to generate a quantized implementation of the Constant Modulus Algorithm (CMA). To quantize the input signal samples, a nearest-element decision device (a slicer), that is typically present in a digital receiver, is used to pre-compute the quantized CMA error function. The number of unique values for the CMA error term is thereby reduced, and the reduced number of CMA error term values are stored in a lookup table. By greatly reducing the number of received signal values used in the CMA error calculation a relatively small lookup table can be used to compute the CMA error function. Passband implementation is accommodated by incorporating the signal de-rotation factor into the lookup table entries. In one embodiment the CMA multiply operation is replaced with shifts and adders. To make efficient use of shift and add operations to achieve multiplication in the adaptation process, the lookup table values are selected to be either a power of 2 or a minimal sum of powers of 2.

FIELD OF THE INVENTION

The present invention relates to equalization techniques to compensatefor channel transmission distortion in digital communication systems. Inparticular, the present invention relates to the efficient baseband andpassband implementations of the Constant Modulus Algorithm (CMA), anequalization algorithm used in blind equalization systems.

BACKGROUND OF THE INVENTION

Digital transmission of information typically involves the modulation ofpulses onto an RF carrier's amplitude and/or phase. Most propagationmediums (terrestrial, cable, underwater, etc.) introduce signaldistortion. Factors that cause distortions include noise, signalstrength variations, phase shift variations, multiple path delays, andthe like.

Noise is also known as static. Signal strength variations are commonlyknown as fading. In addition, multiple different paths between thetransmitter and receiver through the propagation medium cause multiplepath delays. The different paths have different delays that causereplicas of the same signal to arrive at different times at the receiver(like an echo). Multi-path distortion results in inter-symbolinterference (ISI) in which weighted contributions of other symbols areadded to the current symbol.

In addition to distortion and noise from the propagation medium,front-end portions of the receiver and transmitter also introducedistortion and noise. The presence of distortion, noise, fading andmulti-path introduced by the overall communication channel (transmitter,receiver and propagation medium), can cause digital systems to degradeor fail completely when the bit error rate exceeds some threshold andovercomes the error tolerance of the system.

Equalization

Digital systems transmit data as symbols having discrete levels ofamplitude and/or phase. To the extent that a symbol is received at alevel that differs from one of the allowed discrete levels, a measure ofcommunication channel error can be detected.

The digital receiver uses a slicer to make hard decisions as to thevalue of the received signal. A slicer is a decision device responsiveto the received signals at its input, which outputs the projection ofthe nearest symbol value from the grid of constellation points. Theoutput of the slicer thus corresponds to the allowed discrete levels.

At the receiver, it is known to use an equalizer responsive to thedetected error to mitigate the signal corruption introduced by thecommunications channel. It is not uncommon for the equalizer portion ofa receiver integrated circuit to consume half of the integrated circuitarea.

An equalizer is a filter that has the inverse characteristics of thecommunication channel. If the transmission characteristics of thecommunication channel are known or measured, then the equalizationfilter parameters can be set directly. After adjustment of theequalization filter parameters, the received signal is passed throughthe equalizer, which compensates for the non-ideal communication channelby introducing compensating “distortions” into the received signal whichtend to cancel the distortions introduced by the communication channel.

However, in most situations such as in broadcasting, each receiver is ina unique location with respect to the transmitter.

Accordingly, the characteristics of the communication channel are notknown in advance, and may even change with time. In those situationswhere the communication channel is not characterized in advance, orchanges with time, an adaptive equalizer is used. An adaptive equalizerhas variable parameters that are calculated at the receiver. The problemto be solved in an adaptive equalizer is how to adjust the equalizerfilter parameters in order to restore signal quality to a performancelevel that is acceptable by subsequent error correction decoding.

In some adaptive equalization systems, the parameters of theequalization filter are set using a predetermined reference signal (atraining sequence), which is periodically sent from the transmitter tothe receiver. The received training sequence is compared with the knowntraining sequence to derive the parameters of the equalization filter.After several iterations of parameter settings derived from adaptationover successive training sequences, the equalization filter converges toa setting that tends to compensate for the distortion characteristics ofthe communications channel.

In blind equalization systems, the equalizer filter parameters arederived from the received signal itself without using a trainingsequence. In the prior art, it is known to adjust the equalizerparameters blindly using the Least Mean Squares (LMS) algorithm, inwhich the training symbols are replaced with hard decisions, or bestestimates of the original input symbols. Blind equalization systemsusing LMS in this manner are referred to as decision directed LMS(DD-LMS).

However, the DD-LMS algorithm requires a good initial estimate of theinput signal. For most realistic communication channel conditions, thelack of an initial signal estimate results in high decision error rates,which cause the successively calculated equalizer filter parameters tocontinue to fluctuate, rather than converge to a desired solution. Theparameters are said to diverge in such a case.

It is also known to use another algorithm, called the Constant ModulusAlgorithm (CMA), in combination with the DD-LMS algorithm from a coldstart. See D. N. Godard, “Self-recovering equalization and carriertracking in two-dimensional data communication systems,” IEEETransactions on Communications, vol. 28, no 11, pp. 1867-1875, October1980, or J. R. Treichler, B. G. Agee, An New Approach To Muli-PathCorrection Of Constant Modulus Signals, IEEE Transactions On AcousticsSpeech And Signal Processing, vol ASSP-31, no.2, page 459-472 April1983. The CMA algorithm is used first to calculate the equalizer filterparameters, which is regarded as an initial estimate. Thereafter, theequalizer filter parameters (as calculated by the CMA algorithm) areused in an acquisition mode to find the initial equalizer filterparameters to start the DD-LMS algorithm.

The CMA algorithm (as well as the DD-LMS algorithm) is usuallyimplemented with a gradient descent strategy in which the equalizerparameters are adapted by replacing the present equalizer parametersettings with their current values plus an error (or correction) term.See C. R. Johnson, Jr., P. Schniter, T. J. Endres, J. D. Behm, D. R.Brown, R. A. Casas, “Blind equalization using the constant moduluscriterion: a review,” Proceedings of the IEEE, vol. 86, no. 10, pp.1927-1950, October, 1998. The CMA error term itself is a cubic functionof the equalizer output.

From a cold start, the receiver enters an acquisition mode. In theacquisition mode, the CMA algorithm is used first to adjust theequalizer parameters. Then, after a fixed period of time (oralternatively based on a measure, which is derived from the equalizeroutput), the receiver switches to the DD-LMS algorithm in a trackingmode. The acquisition mode typically requires up to 400,000 symbols. Ata 10 MHz clock rate, the symbol rate is 100 nanoseconds and the timeavailable for acquisition using the CMA algorithm is about 40milliseconds. Overall, between the initial CMA algorithm and thefollowing DD-LMS algorithm, the equalizer has about 100-200 millisecondsto converge.

A critical factor in an adaptive equalization system is to complete allthe required multiplication operations within the time available: i.e.,a single symbol interval (100 nanoseconds in the above example). Inparticular, the CMA error term calculation requires successive multiplyoperations for each equalizer parameter. One real multiply per equalizerparameter is needed for one-dimensional signaling, and one complexmultiply (equivalent to 4 real multiplies) per equalizer parameter areneeded for two-dimensional signaling.

Since a typical equalizer filter may have up to 512 filter coefficients(the number of equalizer filter parameters), the total time required tocomplete all the required multiplication operations with full precisionoften exceeds one symbol interval. That is, the large number of multiplyoperations often takes so long that the total time needed forcalculation of all the equalizer filter coefficients exceeds theavailable time limit of one symbol interval. Thus, although the priorart scheme to use CMA and DD-LMS in series is theoretically possible,the large number of multiply operations prevents practical, economicalcommercial implementations in reasonably sized integrated circuitcomponents.

One prior art solution to the problem of economical implementationincludes calculating subsets of the equalizer filter coefficients insuccessive symbol intervals. Another prior art approach is to simplifythe error term multiplication by using only the sign of the error term,i.e., +1 or −1. In LMS, this variation is referred to as signed-errorLMS (SE-LMS) in which the usual LMS error term is replaced by the sign(±1) of the error. SE-LMS is easily implemented since the usualmultiplier per equalizer parameter in the LMS update equation isreplaced by a simple bit flip (a sign change) to representmultiplication by +1 or −1. Similarly, in CMA a signed-error CMA(SE-CMA), is used to replace the usual, full precision CMA error termwith its sign (+1 or −1).

Since multiplication by +1 or −1 is just a sign change, multiplycalculations are very rapid. A modification of the signed error approachis to use three levels, +1, 0, −1 for the error term in themultiplication. Since the number 0 is neither positive nor negative,multiplication by +1, 0, −1 is still a simple and quick operation.However, approximating a term by its sign sacrifices accuracy and canincrease the time required for the adaptive equalizer to converge to asolution. Furthermore, convergence is not guaranteed.

SUMMARY OF THE INVENTION

The present invention is embodied in a blind equalization system inwhich a quantized version of the CMA error function is computed usingquantized input samples and stored in a lookup table (LUT). Inparticular, the quantized CMA error function is pre-computed by firstquantizing input signal samples, then computing the quantized CMA errorfunction based on the quantized input samples and storing the resultingquantized CMA error function in the lookup table memory. Use ofquantized input signal samples to compute the CMA error function may bedescribed as a decision directed CMA (DD-CMA) error function.

To quantize the input signal samples, a nearest-element decision device(a slicer), that is typically present in a digital receiver, isconveniently used in pre-computing the quantized CMA error function.Alternatively, a quantizer with different levels than the availableslicer, and/or more levels than the available slicer is used to quantizethe input samples. The result of quantizing the CMA error function byquantizing the input signal samples increases the quantization precisionin areas of the CMA error function typically encountered when theequalizer parameters are near convergence. Conversely, quantizationprecision is reduced in the areas of the CMA error function typicallyencountered when the equalizer parameters are first set during initialsignal acquisition.

The present technique for computing the DD-CMA error function in placeof the CMA error function reduces the number of unique values for theDD-CMA error term. Due to the relatively reduced number of DD-CMA errorterm values, a lookup table solution becomes practical. That is, theoutput of the slicer assumes relatively few levels (i.e., the correctlevel for each transmitted symbol) as compared to the received signalsamples at the output of the equalization filter, which can assume anylevel. As a result of using the output of the slicer (instead of theinput of the slicer), the DD-CMA error term itself assumes a relativelysmall number of unique values. Each of these possible unique values forthe DD-CMA error term is stored in a lookup table. In such manner, theDD-CMA error term is computed using a relatively small lookup table. Alookup table implementation is easily accomplished with one symbolinterval so that latency is greatly reduced.

For example, in an 8 VSB system using an 8 bit digital/analog converter,the number of possible signal levels at the output of the receiverequalizer is 256. However, at the output of the slicer, number ofpossible signal levels is reduced to 8. By greatly reducing the numberof received signal values used in the DD-CMA error calculation arelatively small lookup table can be used to compute the DD-CMA errorfunction.

In the preferred embodiment the large number of multiply operationsnormally required in the application of the CMA error term for equalizeradaptation is replaced with programmable shifts and adders which valuesare retrieved in the receiver using a lookup table.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a digital communication system employing anequalizer in accordance with the present invention.

FIG. 2 is a block diagram of a baseband equalizer using the CMA and LMSalgorithms in accordance with the present invention.

FIG. 3 is a block diagram of a passband equalizer using the CMA and LMSalgorithms in accordance with the present invention.

FIG. 4 is a block diagram of a decision directed CMA error termcalculator for use in a passband VSB equalizer in accordance with thepresent invention.

FIG. 4A is a block diagram of a decision directed CMA error termcalculator for use in a baseband VSB equalizer in accordance with thepresent invention.

FIGS. 4B and 4C are lookup tables for use in the DD-CMA error calculatorof FIG. 4.

FIG. 4D is a block diagram of a decision directed CMA error termcalculator for use in a baseband QAM equalizer in accordance with thepresent invention.

FIG. 4E is a lookup table for use in the DD-CMA error calculator of FIG.4D.

FIG. 4F is a block diagram of a decision directed CMA error termcalculator for use in a passband QAM equalizer in accordance with thepresent invention.

FIG. 4G is a lookup table for use in the DD-CMA error calculator of FIG.4F.

FIG. 5 is a graph of a CMA error function showing the use of 8 mid-treadquantization levels to quantize the DD-CMA error function in accordancewith the present invention.

FIG. 6 is a graph of a CMA error function showing the use of 16mid-tread quantization levels to quantize the DD-CMA error function inaccordance with the present invention.

FIG. 7 is a graph of a CMA error function showing the use of 16 mid-risequantization levels to quantize the CMA error function in accordancewith the present invention.

FIG. 8 is a graph of the in-phase component of the CMA error term for64-QAM signaling shown as a surface above the I-Q plane with thein-phase component of the CMA error term being a function of twovariables (I and Q).

FIG. 9 is a graph of the in-phase component of the DD-CMA error term for64-QAM signaling with 8-level slicing, shown as a surface above the I-Qplane, with the in-phase component of the DD-CMA error term being afunction of two variables (I and Q).

FIG. 10 is a graph of the in-phase component of the DD-CMA error termfor 64-QAM signaling with 16-level slicing, shown as a surface above theI-Q plane with the in-phase component of the DD-CMA error term being afunction of two variables.

FIG. 11 is a graph of the in-phase component of a passband CMA errorterm for 8-VSB signaling used in conjunction with the present invention.

FIG. 12 is a graph of the in-phase component of a CMA error function forQAM signaling used in conjunction with the present invention.

DETAILED DESCRIPTION

A typical quadrature amplitude modulation (QAM) communications systemhas a transmitter station 10 and a receiver station 14, coupled togethervia a suitable propagation medium 12, as shown in FIG. 1. Thetransmitter station 10 includes an information source 16 such as video,audio and/or data coupled to a digital encoding subsystem 18. Thein-phase component (I) and quadrature phase component (Q) from the QAMdigital encoding subsystem 18 are coupled to QAM modulator 20, whichmodulates the I and Q signal components onto a suitable carrierfrequency (provided by carrier oscillator 22) for transmission into thepropagation medium 12.

The receiving station 14 includes a tuner 24, demodulator, A/Dconverter, and AGC (automatic gain control) functions 26, and a timingrecovery module 28. The timing recover module 28, reproduces the signaltime slicing so that it is aligned with the original I and Q signals inthe receiver. The receiving station 14 further includes an adaptiveequalizer 30, a carrier recovery function 32, an error correctiondecoding function 34, and a digital to analog (D/A) converter 36. All ofthe elements between the original I and Q signals at the output of thedigital encoding subsystem 18 in the transmitter, up to the input to theequalizer 30 in the receiver is regarded as the overall communicationchannel 13. The function of the equalizer 30 in the receiver is tocompensate for distortion and noise originating anywhere in the overallcommunication channel 13.

In operation, the tuner 24 selects an appropriate carrier frequency fromthe propagation medium 12. The output of tuner 24 is converted todigital samples in an A/D converter and demodulated 26 to the correctfrequency range. Also, the AGC 26 feedback loop automatically adjuststhe receiver gain level. The timing recovery function 28 pulls the localcrystal oscillator which governs the A/D sampling clock into phase lockwith the incoming signal. The recovered I and Q signal components fromthe timing recovery module 28 are input to the equalizer 30.

The output of the equalizer 30 is coupled to a carrier recovery module32 which pulls the local crystal oscillator into precise carrierfrequency and phase lock, and provides data bit estimates to the errorcorrection decoder 34. After error correction decoding 34, the digitaldata is recovered, and reproductions of the original data, video, andaudio are output from the receiver 14. The present invention is embodiedin the equalizer 30 portion of the communication system.

Passband Forward Equalization

Further details of the equalizer 30 and carrier recovery 32 portions ofFIG. 1 are shown in FIG. 2. As indicated, the overall communicationchannel 13 introduces system distortion 41 and noise 43 into the I and Qsignal components from the transmitter. The received I and Q signalcomponents at the receiver are input to the forward equalizer 40, whichis typically implemented as a finite impulse response (FIR) filter. Theoutput of the forward equalizer 40 is input to a mixer (multiplier) 48which translates the processed I and Q signal components to basebandfrequency (DC). The process of translating the received signal tobaseband (demodulation), is also called de-rotation. See Lee andMesserschmitt, Digital Communications, Kluwer Academic Publishers, 1988.

The output of the mixer 48 is coupled to a slicer 50, which is set fornominal decision threshold levels that correspond to each symbol in theexpected QAM constellation. The input YI and YQ to the slicer 50 are theactual received signal levels which have been filtered 40 and de-rotated48. After the slicer 50, the output YI{circumflex over ( )} andYQ{circumflex over ( )} represent hard decision levels which correspondto the expected signal levels in the QAM constellation.

The phase detector 46, loop filter 44 and sine/cosine generator 42 incombination with multiplier 48 constitute a closed loop for recoveringthe frequency and phase of the carrier signal. The phase difference ofinput signals 49 (YI and YQ) and output signals 52 (YI{circumflex over ()} and YQ{circumflex over ( )}) of slicer 50 is detected in phasedetector 46. The detected phase difference, filtered in loop filter 44controls the frequency and phase of the sine/cosine generator 42 in adirection so as to reduce the detected phase difference between theinput signals to the phase detector 46. At steady state, the carrierloop tracks the input carrier frequency and phase. See Meyr et al.,Digital Communication Receivers, John Wiley & Sons, 1998.

Both input YI and YQ and output YI{circumflex over ( )} andYQ{circumflex over ( )} are input to an error term calculator 54. To theextent that the inputs and outputs of the slicer 50 are not equal, theerror term calculator 54 provides an output to update the passband errorterm in the forward equalizer 40. The error term calculator operates onbaseband signals. In order to generate a passband error term, the localcarrier loop signal (at the output of sine/cosine generator 42) is inputto the error term calculator 54. The present invention relates to theefficient implementation of the error term calculator 54 which alsoleads to an efficient implementation of the equalizer update in theforward equalizer, 40.

Baseband Feedback Equalization and Passband Forward Equalization

An alternate implementation of the equalizer 30 and carrier recovery 32of FIG. 1, is shown in FIG. 3. The implementation of FIG. 3 is similarto that of FIG. 2, except that feedback equalization at baseband isincluded in addition to forward equalization at passband. The addedelements in FIG. 3 are an adder 66 and a feedback equalizer 74. Theforward equalizer 60 is substantially the same as the forward equalizer40 in FIG. 2. The slicer 68 in FIG. 3 is substantially similar to theslicer 50 of FIG. 2. Carrier loop 64 in FIG. 3 encompasses substantiallythe same components as sine/cosine generator 42, loop filter 44 andphase detector 46 of FIG. 2. However, while error calculator 72 in FIG.3 performs all the functions of error term calculator 54 in FIG. 2, itfurther provides an output signal to update the baseband error term tothe feedback equalizer 74. The feedback equalizer 74 is implementedusing an FIR filter imbedded in a feedback loop which makes the overallloop have an infinite impulse response.

Both inputs to the slicer YI and YQ, and outputs of the slicerYI{circumflex over ( )} and YQ{circumflex over ( )} are input to theerror term calculator 72. To the extent that the inputs and outputs ofthe slicer 68 are not equal, the error term calculator 72 provides anoutput to update the passband error term in the forward equalizer 60. Asbefore, in order to generate a passband error term, the local carrierloop signal is input to the error term calculator 72.

Error term calculator 72 provides error term updates for the parametersof both forward equalizer 60 at passband and feedback equalizer 74 atbaseband. The present invention relates to the efficient implementationof the error term calculator 72, which also leads to an efficientimplementation of the equalizer update in the forward equalizer, 60, andthe feedback equalizer 74.

As indicated above, the equalizer parameter vector is updated accordingto a gradient descent strategy in which the average parameter trajectoryfollows the steepest slope of the specified cost surface (i.e., theparameters move on average in the direction of the derivative of thecost function).

The Constant Modulus cost function is given by

J _(CM) =E{(g ² −|y| ²)²}

where g² is a scalar referred to as Godard's (dispersion) constant, y isthe baseband equalizer output, and E{.} denotes expectation. Thegradient can be found using complex vector calculus and is described byGodard or Treichler, cited above.

The baseband CMA error term e_(bb)(k) is the derivative of J_(CM) withthe expectation removed and is given by

e _(bb)(k)=y(k)(g ² −|y(k)|²)

The CMA error term e_(bb)(k) is applied to a regressor vector of inputsamples which is of length equal to the number of equalizer parametersso that one multiplication per equalizer parameter is required to updatethe equalizer tapped delay line. For high data rate signaling, thenumber of multiplications can become computationally prohibitive.Reducing the computational burden is the motivation for the signedalgorithms of the prior art described above. A sign operation(multiplication by 0, +1 or −1) is easily implemented as a bit flip intwo's complement arithmetic. Therefore, no multiplier is needed, whichsignificantly reduces the required chip die area. The present inventionuses a quantized version of the CMA error term, which is implemented inlookup tables, and achieves low-precision multiplication using bitshifts and adders.

In accordance with the present invention, the performance of the CMAalgorithm in a blind equalization system is improved by quantizing theCMA error function by using quantized input samples to compute theDD-CMA error function and storing the resulting quantized DD-CMAfunction in one or more lookup tables. Block diagrams of various DD-CMAerror calculators (quantizers) embodying the present invention are shownin FIGS. 4-4G. The baseband DD-CMA error term quantizer of FIG. 4D maybe used in the baseband CMA error term calculator 72 in FIG. 3. Thepassband DD-CMA error term quantizer of FIG. 4F may be used in CMA errorterm calculator 72 in FIG. 3 and 54 in FIG. 2.

The quantizer of the present invention uses a variety of quantizationstrategies for efficient implementation of the Constant ModulusAlgorithm (CMA) . Each of the various strategies is based on quantizingthe standard update CMA error term by quantizing the input samples,calculating the DD-CMA error term and storing the resulting quantizedDD-CMA error term in a plurality of lookup tables. Thereafter, theDD-CMA error term can be efficiently determined during the receiverequalization process by accessing the appropriate lookup table. Theinvented decision based quantization is applied to both one-dimensionalsignals (such as Vestigial Sideband Modulation (VSB)) andtwo-dimensional signals (such as Quadrature Amplitude Modulation (QAM))for both baseband and passband implementations.

FIG. 4A shows a lookup table implementation of a DD-CMA error termcalculator for use in a baseband VSB system. The received signal Y atthe output of the equalizer, is input to slicer 401 which selects thenearest expected signal level, Y{circumflex over ( )}. The slicer outputY{circumflex over ( )} is used as an input address to lookup table 400from which the stored pre-computed value of the quantized DD-CMA errorterm, e_(bb), corresponding to the input signal Y is retrieved.

FIG. 4 shows a lookup table implementation of a DD-CMA error termcalculator for use in a passband VSB system. In FIG. 4, the input Y, atinput terminal 480 is the baseband equalizer output. The sine and cosinewaveforms in digital form corresponding to the local oscillator of thereceiver are available as inputs. Also provided is a module 411 forcomputing y_(cos)(k), and y_(sin)(k), as defined below. Slicers 417 and419 quantize each respective input to determine hard decisions. ThoughFIG. 4 shows two separate slicers, it is understood that access to asingle on-chip slicer will be time-shared. The error quantizer of FIG. 4includes an error function selector 410 responsive to the inputsine/cosine input, to select an error function curve. Selector 410 alsoselects a lookup table from a plurality of lookup tables 416, 418, 420(LUT 1, LUT 2, and LUT N), one for each possible selected error functioncurve. N in FIG. 4 is at most equal to the number of angle used in thesine/cosine generation. Multiplexers (MUX) 414 and 422 are responsive toselector 410, to select the appropriate lookup table 416, 418, or 420.Multiplexers 414 and 422 are implemented in digital form such as bycomputing the address where a desired lookup table is located in memory.The selected lookup table output provides the computed value of thequantized DD-CMA error term e_(bb)(k) at terminal 498.

The CMA error term as shown by the solid curve in FIG. 5, is a cubicfunction of y which crosses through the origin, with roots y={0,±g}. TheCMA error calculation requires two multiplies and one add for realsignal processing. In high data rate scenarios, the computation of theerror term and its application to the update of the equalizer parameterscan induce significant time delay that exceeds the symbol period. Thisprocessing delay can seriously degrade performance. The goal is theefficient approximation of the CMA error term, which is easilyimplemented within a symbol period.

A digital receiver chip normally contains a non-linear nearest-elementdecision device in hardware. The existing decision device takes signalsample y as its input and outputs the constellation point (or symbolvalue) y{circumflex over ( )} which has closest Euclidean distance tothe input. The decision boundaries are chosen midway between symbolvalues.

In the present invention, the DD-CMA error term is computed usingquantized input estimates, y{circumflex over ( )}, instead of fullprecision samples Y. For example the DD=CMA error term for baseband VSBsignaling is given by

e(DD-CMA)=Y{circumflex over ( )}(g ² −|y{circumflex over ( )}| ²)

As a consequence, the DD-CMA error function assumes a finite number ofvalues since y{circumflex over ( )} assumes a finite number of values.For example, with M-VSB signaling, the decision directed CMA (DD-CMA)error term assumes at most M/2 unique magnitudes. Hence, the basebandDD-CMA error term is efficiently calculated by using a lookup table 400which is addressed using y(k), as illustrated in FIG. 4A. The lookuptable 400 in FIG. 4A replaces the two multiplies and one add needed tocalculate the usual CMA error term, and has no greater than M possibleentries (M/2 unique magnitudes with 2 sign polarities). As indicatedabove, the lookup tables are configurable to include VSB and QAM in bothbaseband and passband implementations.

FIGS. 4B, 4C, 4E and 4G illustrate various implementations of lookuptables that may be used for the lookup tables (416, 418, 420) in FIGS.4, 4D and 4F. In particular, FIGS. 4B and 4C show the lookup tableimplementations with double indexing for the I and Q signal components,respectively, of the DD-CMA error term for passband VSB signaling inFIG. 4. FIG. 4E shows the lookup table implementation with doubleindexing for I and Q signal components, respectively, of the DD-CMAerror term for baseband QAM signaling in FIG. 4D. FIG. 4G shows thelookup table implementation with double indexing for I and Q signalcomponents, respectively, of the DD-CMA error term for passband QAMsignaling 4F.

VSB Signaling

As previously described the baseband CMA error term e_(bb)(k) forbaseband signaling is given above as a cubic function of the equalizeroutput Y. The effect of quantizing the CMA error term is illustrated inFIG. 5. The solid line in FIG. 5 is the true CMA error term for 8-VSBsignaling. The dashed line in FIG. 5 is the decision directed CMA(DD-CMA) error function computed in accordance with the presentinvention. The DD-CMA error term has six unique levels and three uniquemagnitudes. (By coincidence in FIG. 5, the magnitude of 84 is used twiceon each side of the origin). For large-magnitude equalizer samples, FIG.5 shows only one quantization level (about ±84). Having only a singlequantization level for large output samples may slow initial algorithmconvergence.

Extending the Slicer Values

Extending the slice levels of the decision device can substantiallyreduce the quantization error. For example, suppose the decision deviceis extended to slice 16-level in lieu of 8-level VSB signals. The 8-VSBlevels are denoted by {±2, ±6, ±10, ±14}. The 16-level slice points aredenoted by {±1, ±3, ±5, ±7 ±9, ±11, ±13, ±15}. By quantizing the 8-VSBsignal as a 16-VSB signal, the quantization error is substantiallyreduced, as shown by the quantized DD-CMA error function in FIG. 6.Furthermore, the quantized DD-CMA error term (dashed line in FIG. 6)assumes two levels (approximately ±273 and ±1155) for larger equalizeroutputs, as opposed to one quantization level (about ±84) when an8-level slicer is used (dashed line in FIG. 5). Having multiplequantization levels for larger equalizer outputs tends to improve theinitial algorithm convergence rate. The extension of the decision deviceto include 16-level over 8-level slicing requires only a minimalincrease in combinatorial logic. The quantization error, of the CMAerror term, however, is substantially improved.

Mid-Tread Quantization

In each of the algorithms discussed above, the DD-CMA error term hasnon-zero values near the roots of the true CMA error term, i.e. at theorigin and y=±g. These non-zero values of the DD-CMA error term tend tointroduce jitter as the equalizer parameters converge towards asolution. The latter type of quantization is called mid-risequantization, as compared to a mid-tread quantization, which assumes azero value in some region on either side of the root locations. Since amid-tread quantizer contains the zero value as a valid output, it mapssmall positive or negative values of CMA error to zero. Also, since thetrue CMA error is zero at the root locations, a mid-tread quantizer canhave significantly better performance over a mid-rise quantizer. Thedifferences between mid-tread and mid-rise quantizers are discussed byN. S. Jayant and P. Noll, in “Digital coding of waveforms,” EnglewoodCliffs, N.J.: Prentice Hall, 1984.

In an alternate embodiment of the present invention, a mid-risequantization is converted to mid-tread quantization by modifying theslice levels of the decision device. A bias is applied to the slicevalues to shift the decision levels. For example, suppose the slicelevels are chosen as {0, ±2, ±4, ±6 ±8, ±10, ±12, ±14} instead of {±1,±3, ±5, ±7 ±9, ±11, ±13, ±15}. In such case, one half the magnitude ofthe minimum symbol is subtracted from the magnitude of each slice point.The bias removal yields mid-tread slice levels. The 8-VSB symbol valuesare now equal to every other 16-level slice value. Decision boundariesare chosen midway between slice points.

FIG. 7 shows the resulting DD-CMA error term using mid-treadquantization (dashed line) for 8-VSB signaling. The mid-tread symbolquantization of the input symbols manifests itself in the CMAquantization as one mid-tread quantization at the origin, and twonear-mid-tread CMA quantization levels at the cubic roots. As shown inFIG. 7, the DD-CMA error term assumes the zero value at the origin, butis still non-zero at the roots at ±g.

The reason that the DD-CMA error term value is not exactly zero near the±g roots is that the Godard radius is not exactly equal to the square ofthe slice level near y=±g. For example, g²=148 for 8-VSB signaling withthe symbol set {±2, ±6, ±10, ±14}. However, the 16-level decision deviceproduces a value of Y{circumflex over ( )}=12 for 11<y<=13, and 12²=144which does not exactly equal g² (148). By changing the Godard radius (to144 in this case) the DD-CMA error term assumes zero values near allroots of the cubic. Alternatively, the quantization value at the rootlocations can be manually set to zero. A zero-state at the cubic rootshelps reduce the mean-squared error or stochastic jitter as theequalizer converges to a solution.

Passband Error

For passband operation, (away from DC) the CMA error term is a rotatedversion of the baseband CMA error term; i.e., e_(pb)=e^(jθ(k))e_(bb).The in phase (I) and quadrature (Q) components of the passband CMA errorterm are given separately as,

e _(pb) I=cos (θ)*y(k)(g ² −|y(k)|²)

e _(pb) Q=sin (θ)*y(k)(g ² −|y(k)|²)

For VSB signaling where information is encoded in amplitude only (andnot phase), y(k) is real-valued. In such case, the I and Q components ofthe passband error term are a function of the rotation angle and theequalizer output. The passband error components are scaled-in-magnitudeversions of the baseband error term, but the roots of the cubic errorfunction remain the same as the baseband error function. FIG. 11 showsthe in-phase component of the passband CMA error term versus equalizeroutput Y for rotation angles 0, 30, 45 and 70 degrees.

A quantization procedure (similar to that used for the baseband errorterm) that will “track” the scaling due to rotation, is desired. Thepassband samples, defined as y_(cos)(k)=cos(θ)*y(k) andY_(sin)(k)=sin(θ)*y(k) are passed into the nearest-element decisiondevice and yield samples y_(cos)(k){circumflex over ( )} andY_(sin)(k){circumflex over ( )}, respectively. A DD-CMA error term forpassband implementation is analogous to that which was disclosed forbaseband implementation.

e _(pb) I(DD-CMA)=[y _(cos)(k){circumflex over ( )}](g ²−|y(k){circumflex over ( )}|²)

e _(pb) Q(DD-CMA)=[y _(sin)(k){circumflex over ( )}](g ²−|y(k){circumflex over ( )}|²)

The lookup tables in FIGS. 4B and 4C are indexed (addressed) toimplement the above equations, respectively. The look-up table must beaddressed by y_(cos)(k){circumflex over ( )} and y(k){circumflex over ()} to calculate the I-component of the DD-CMA error term, ory_(sin)(k){circumflex over ( )} and y(k){circumflex over ( )} tocalculate the Q-component, as illustrated in FIGS. 4B and 4Crespectively. In particular, the lookup table for the passbandimplementation of the I component of the CMA error term is shown in FIG.4B. The lookup table for the passband implementation of the Q componentof the CMA error term is shown in FIG. 4C.

QAM Signaling

QAM signaling encodes information into both the RF carrier's amplitudeand phase, and has become the standard for digital cable televisiontransmission. In this case, the baseband equalizer output, y, iscomplex. Let y be represented by its real and imaginary parts, y=I+jQ.The baseband CMA error term can be written as

 e _(bb)=(I+jQ)*(g ² −I ² −Q ²)

which is separated into in-phase and quadrature components as follows,

e _(bb) I=I*(g ² −I ² −Q ²)

e _(bb) Q=Q*(g ² −I ² −Q ²)

Unlike VSB signaling, there are two baseband components of the CMA errorterm. Let I{circumflex over ( )} and Q{circumflex over ( )} be theoutput of the nearest-element decision device for inputs I and Q,respectively. The analogous DD-CMA to the real-signaling case is toreplace I by I{circumflex over ( )} and Q by Q{circumflex over ( )} ineach of the in-phase and quadrature components of the error term, or

e _(bb) I(DD-CMA)=I{circumflex over ( )}*(g ² −I{circumflex over ( )} ²−Q{circumflex over ( )} ²)

e _(bb) Q(DD-CMA)=Q{circumflex over ( )}*(g ² −I{circumflex over ( )} ²−Q{circumflex over ( )} ²)

For 64-QAM signaling with I and Q symbol values chosen from {±1, ±3, ±5,±7}, the in-phase component of the CMA error term is a function of thetwo variables, I and Q. The in-phase component is illustrated as asurface above the I-Q plane in FIG. 8. After quantization in accordancewith the present invention, the continuous shape is quantized intodiscrete levels. FIG. 9 shows the I-component of the DD-CMA error termas a quantized surface above the I-Q plane, using 8-level slice valueschosen from {±1, ±3, ±5, ±7}. For M² QAM signaling, there are at mostM²/2 levels in each of the I and Q components of the DD-CMA error term.

The quantization in FIG. 9 is coarse, but may be adequate in manyapplications. Significant improvement can be found by using a 256-QAM(16-levels in I and Q components) decision device for 64-QAM signaling.(This approach is similar to that used above for 8-VSB signaling byextending the decision device to include 16-VSB slice values). Forexample in 64-QAM, the I and Q symbol values are chosen from {±2, ±6,±10, ±14} and the slice points are chosen as {±1 ±3 . . . ±15}. FIG. 10shows the quantized DD-CMA error term as a surface above the I-Q plane.A 256-QAM 16-level slicer is used on 64-QAM samples to produce thedecisions for the DD-CMA error term. The addition in complexity to thedecision device is minimal, while the quantization of the update CMAerror term is substantially improved.

Passband Implementation

When the equalizer processes samples which are away from DC (passband),the baseband error term is rotated to form the passband error term, ore_(pb)=e_(bb)e^(jθ(k)). The I and Q components of the passband CMA errorterm are given by

e _(pb) I=[cos (θ)I−sin (θ)Q]*(g ² −I ² −Q ²)

e _(pb) Q=[cos (θ)Q+sin (θ)I]*(g ² −I ² −Q ²)

The components of the passband equalizer output sample are defined asI_(pb)=[cos(θ)I−sin(θ)Q] and Q_(pb)=[cos(θ)Q+sin(θ)I] and the quantizedvalues from the decision device are I_(pb){circumflex over ( )} andQ_(pb){circumflex over ( )}, respectively. The decision directed CMA(DD-CMA) update error term is found by replacing the baseband andpassband samples in the usual passband CMA error term with their bestestimates from the decision device, or

e _(pb) I(DD-CMA)=I _(pb){circumflex over ( )}*(g ² −I{circumflex over ()} ² −Q{circumflex over ( )} ²)

e _(pb) Q(DD-CMA)=Q _(pb){circumflex over ( )}*(g ² −I{circumflex over ()} ² −Q{circumflex over ( )} ²)

To implement the above equations in lookup tables would require tripleindexing (addressing). That is, each component of the in-phase andquadrature component of the DD-CMA error term is a function of threevariables: the quantized baseband samples and each of the components ofthe quantized I and Q passband samples.

To use look-up tables to calculate the DD-CMA error term addressed byI_(pb){circumflex over ( )}(k), I{circumflex over ( )}(k), andQ{circumflex over ( )}(k) (for the I-component of the DD-CMA errorterm), or addressed by Q_(pb){circumflex over ( )}(k), I{circumflex over( )}(k), and Q{circumflex over ( )}(k) (for the Q-component of theDD-CMA error term) would require triple indexing.

Double Indexing of the Lookup Tables

The DD-CMA error term can be modified for QAM passband signaling so thatthe requirement of triple indexing is reduced to double indexing of thelook up table, which greatly reduces the memory size of the requiredlookup table. For example, since the passband sample is a rotatedversion of the baseband sample, then I² ₊Q²=I_(pb) ² ₊Q_(pb) ², i.e.,the passband and baseband samples have equal squared magnitudes. Whenquantized samples are considered, I{circumflex over ( )}² ₊Q{circumflexover ( )}² is not in general equal to I_(pb){circumflex over ( )}hu 2₊Q_(pb){circumflex over ( )}². However, if it is assumed thatI{circumflex over ( )}² ₊Q{circumflex over ( )}² can be approximated byI_(pb){circumflex over ( )}² ₊Q_(pb){circumflex over ( )}², then thein-phase and quadrature components of the DD-CMA error terms forpassband QAM signaling are modified to be

e _(pb) I(DD-CMA)=I _(pb){circumflex over ( )}(g ² −I _(pb){circumflexover ( )}² −Q _(pb){circumflex over ( )}²⁾

and

e _(pb) Q(DD-CMA)=Q _(pb){circumflex over ( )}(g ² −I _(pb){circumflexover ( )}² −Q _(pb){circumflex over ( )}²)

For the above equations, the lookup tables in FIG. 4F require doubleindexing, as shown in FIG. 4G, instead of triple indexing.

The number (N) of lookup tables (LUT 1 to LUT N in FIG. 4) required forimplementation of the present DD-CMA algorithm depends upon themodulation scheme used at the transmitter, and whether baseband orpassband processing is used at the receiver. For example, FIG. 11illustrates the in-phase component of a passband CMA error term for8-VSB signaling. The I component of the passband CMA error term for8-VSB signaling is shown at rotation angles of 0 (baseband), 30, 45 and70 degrees. In VSB, there is essentially one CMA error curve, which ismultiplied by a rotation angle factor to obtain the CMA error term.Therefore, only one lookup table (FIG. 4A) is required for VSB basebandprocessing. For VSB passband processing, the size of the lookup table isenlarged to accommodate indexing as a function of the rotation angle.

In the case of quadrature amplitude modulation (QAM) there are a familyof CMA error curves. The in-phase component of a CMA error function forQAM signaling is shown in FIG. 12. For QAM signaling, where informationis encoded into the RF carrier in both amplitude and phase, the basebandequalizer output, y, is a complex variable. The desired CMA errorfunction is then determined from a family of CMA error function curvesrather than the single baseband error function curve, as in the case ofVSB signaling in FIG. 11.

In the general case (including QAM signaling), each of the family of CMAerror function curves is independently quantized using theabove-described DD-CMA approach. The pre-computed quantized CMA errorfunction curves are stored in a corresponding plurality of lookuptables.

After the lookup tables are created, the receiver is ready to processdigital signals. During QAM signal acquisition, when the equalizerparameters are being initialized, the appropriate CMA error functioncurve (selected from the family of CMA error function curves in FIG. 12)is determined. The lookup table corresponding to the selected CMA errorfunction curve is selected at the receiver. The selected lookup table,which stores the pre-computed DD-CMA error function (e_(n)), isretrieved and used to adjust the parameters of the adaptive equalizer.In an M² level-QAM system, the number of unique error function curves isequal to (M/2). By way of example, in a 64-QAM system, where M=8, 4lookup tables are required for baseband signal processing. For QAMpassband processing, the size of the lookup tables is enlarged toaccommodate indexing as a function of the rotation angle.

Baseband VSB Signaling, FIG. 4A

In operation in FIG. 4A, the DD-CMA error term is determined byaddressing a lookup table 400, by the single variable, Y{circumflex over( )} which is a quantized 401 version of the received signal Y at theoutput of the equalizer. If N-level signaling is used with symbol values{−(N−1) . . . ±3, ±1, 1, 3, . . . (N−1)}, the DD-CMA error term assumesN/2 magnitudes for elements of this symbol set. The pre-computedelements are stored in the lookup table, and retrieved from the lookuptable 400 by the index (address) of the value of Y{circumflex over ( )}in the symbol set.

Baseband VSB Signaling, FIG. 4

In operation in FIG. 4, the baseband equalizer output at input terminal480 is applied to computing module 411. The cosine (or sine) of thereceived carrier phase angle _(θ) at terminal 481 is applied to thelookup table selector 410. For each of the possible received carrierphase angles, _(θ), a CMA error function curve (from a family of curvessuch as shown in FIG. 11) is selected 410. The cosine (or sine) of thereceived carrier phase angle is also applied to computing module 411where the respective product term, y_(cos)(k) (or Y_(sin)(k)), iscalculated.

The calculated product term(s) are quantized into discrete levels inslicer 417, as is the input Y in slicer 419. The N lookup tables 416,418, 420 are thus addressed by two variables: y_(cos)(k){circumflex over( )} (or Y_(sin)(k){circumflex over ( )}) and Y{circumflex over ( )}.Variables y_(cos)(k){circumflex over ( )} and Y{circumflex over ( )}address the bank of look up tables for retrieval of the I component ofthe DD-CMA error term, while variables Y_(sin)(k){circumflex over ( )}and Y{circumflex over ( )} address the bank of look up tables forretrieval of the Q component of the DD-CMA error term. The I and Qcomponents of the DD-CMA error term are to be retrieved from among Nlookup tables, where the number of lookup tables is equal to the numberof quantization levels of the received carrier phase angle, _(θ).

For each of the possible DD-CMA error curves that might be selected 410,an appropriate lookup table is determined. Based on the selected DD-CMAerror function curve, multiplexer 414, responsive to the selectionmodule 410, applies the quantized terms from slicers 419, 417, i.e.,Y{circumflex over ( )} and Y_(cos)(k){circumflex over ( )} (orY_(sin)(k){circumflex over ( )}) to the selected lookup table. At thesame time, multiplexer 422, responsive to the selection module 410,selects the output of the appropriate lookup table 416, 418, 420 (LUT 1,LUT 2, and LUT N) to go to the output terminal 498.

The lookup tables for use with the DD-CMA error quantizer of FIG. 4 areshown in FIGS. 4B and 4C. To compute the I component of the DD-CMA errorterm, the lookup table in FIG. 4B in used in FIG. 4. To compute the Qcomponent of the DD-CMA error term, the lookup table in FIG. 4C in usedin FIG. 4. The output of the selected lookup table at terminal 498 isthe quantized value of the DD-CMA error term, e_(bb)(k) , which is usedin further computations to the update the parameters of the equalizationfilter.

Thus the error quantizer of FIG. 4 uses hard decision values to quantizethe CMA error function, which hard decision values are used topre-compute DD-CMA error function values which are stored in a pluralityof lookup tables, and retrieved during the equalization process. Otherembodiments of the invention are shown in the examples in FIGS. 4D and4F.

Baseband QAM, FIGS. 4D, 4E

The operation of the CMA error term quantizer in FIG. 4D is similar tothat of FIG. 4. In operation in FIG. 4D, the input I{circumflex over ()} and Q{circumflex over ( )} values are applied to the lookup tableselector 510. For each pair of expected values of I{circumflex over ( )}and Q{circumflex over ( )}, a CMA error function curve is selected 510,from a family of CMA error function curves (as for example, as shown inFIG. 12).

The input I{circumflex over ( )} and Q{circumflex over ( )} values arealso applied to multiplexer 514. For each of the possible CMA errorcurves that might be selected 510, an appropriate lookup table 516, 518or 520 is determined. Based on the selected CMA error function curve,multiplexer 514, responsive to the selection module 510, applies thequantized input terms I{circumflex over ( )} and Q{circumflex over ( )}to the selected lookup table via multiplexer 514. At the same time,multiplexer 422, responsive to the selection module 410, selects theoutput of the appropriate lookup table 416, 418, 420 (LUT 1, LUT 2, andLUT N) to go to the output terminal 498.

A lookup table for use with the CMA error quantizer of FIG. 4D is shownin FIG. 4E. For baseband M²-QAM signaling, the I and Q components of theDD-CMA error term require M/2 lookup tables(for example, M=8 in 64-QAMwhich requires 4 lookup tables). The indices of I{circumflex over ( )}and Q{circumflex over ( )} address the bank of lookup tables and theDD-CMA error term is retrieved at the output terminal of the multiplexer522. When computing the I component of the DD-CMA error term, the indexof the value of Q{circumflex over ( )} controls the multiplexers 514,522. Similarly, when computing the Q component of the DD-CMA error term,the index of the value of I{circumflex over ( )} controls themultiplexers 514, 522.

The output of the selected lookup table at terminal 598 provides thequantized I and Q components of the CMA error values of the CMA errorterm, e_(bb)(k) , which are used in further computations to the updatethe parameters of the equalization filter.

Passband QAM, FIGS. 4F and 4G

For passband QAM signaling, an approximation is made that I_(pb)²+Q_(pb) ² is approximately equal to I_(pb){circumflex over ()}²+Q_(pb){circumflex over ( )}². As a result of the approximation, twovariables rather than three variables are needed to address the bank oflookup tables. FIGS. 4F and 4G show the implementation for the retrievalof the I and Q components of the DD-CMA error term.

The operation of the DD-CMA error term quantizer in FIG. 4F is similarto that of FIGS. 4 and 4D. In operation in FIG. 4F, the input I and Qvalues are applied to computing module 611. At the same time, the sineand cosine of the received carrier phase angle _(θ) is applied atterminal 681 and input to computing module 611, which computes theformulas indicated. The output of the computing module 611, labeledI_(pb) and Q_(pb), are coupled to slicers 617 and 619, respectively, toprovide their expected values, I_(pb){circumflex over ( )} andQ_(pb){circumflex over ( )}, which are applied to selector module 610and the inputs to multiplexer 614.

For each pair of possible expected values I_(pb){circumflex over ( )}and Q_(pb){circumflex over ( )}, one of the DD-CMA error function curves(as for example, the family of error function curves shown in FIG. 12)is selected 610. For each of the possible DD-CMA error curves that mightbe selected 610, an appropriate lookup table is determined.

Based on the selected CMA error function curve, multiplexer 614,responsive to the selection module 610, applies the quantized terms fromslicers 617, 619, i.e., I_(pb){circumflex over ( )} andQ_(pb){circumflex over ( )}, to the selected lookup table. At the sametime, multiplexer 622, responsive to the selection module 610, selectsthe output of the appropriate lookup table 616, 618, 620 (LUT 1, LUT 2,and LUT N) to go to the output terminal 698. The lookup tables for usewith the DD-CMA error quantizer of FIG. 4F are shown in FIG. 4G. Theoutput of the selected lookup table at terminal 698 is the quantizedvalue of the DD-CMA error term, e_(bb)(k), which is used in furthercomputations to the update the parameters of the equalization filter.

Low Complexity Implementation

The above described methods for producing low-complexity versions of theCMA error term for baseband and passband equalizers include quantizationof the update DD-CMA error term to assume a finite number of values,suitable for using a lookup table implementation, and thus reducingmultiplier count. In addition, the delay in processing the updateequation is reduced.

After the lookup table, the update CMA error term must still be appliedto each equalizer parameter in order to update the entire tapped delayline. One implementation for the application of the error term is to uselow bit-width multipliers. For example, in the absence of our DD-CMAerror term, typically 12-16 bit precision is used to represent theequalizer error term. If the data in the regressor vector is stored with10-bit precision, this implies the use of a (12-16×10) multiplier foreach equalizer parameter. However, with the low precision of the DD-CMAerror term, 4-6 bit precision is sufficient and multiplier area issignificantly reduced.

Alternatively, shifts and adders can be used in place of multipliers.For this purpose, contents of the lookup tables (the DD-CMA error termvalues) are selected to be powers of 2, or the sum of powers of 2, whichnumbers make efficient use of shift and add operations to achievemultiplication in the adaptation process.

For example, use a canonical-sum-digits representation to write theerror term as

e(DD-CMA)=Σ_(I)(a _(I)*2^(−Zi))

where a_(I) assumes values {0,−1,+1}, and Z_(I) are integers. Theproduct of the regressor sample, r(k), and the error term is thenexpressed as

e(DD-CMA)=Σ_(I)(a _(I) *r(k)*2^(−Zi))

which can be implemented using shifts and adders. Since the error termis generated using a look-up table, the look-up values can be changed tooutput the a, values instead. In the general case, addition includessubtraction when signed factors are considered. Numerical examplescorresponding to FIGS. 5, 6 and 7 are given below.

As labeled in FIG. 5, the quantized CMA error term for 8-VSB signalinghas six unique levels and three unique magnitudes (84 is used twice oneach side of the origin by coincidence). Each of these quantizationlevels are factored into powers of 2 as follows:

84=64+16+4

60=64−4

36=32+4

Note that the use of negative numbers for the powers of 2 summationresults in fewer shift and add operations. For example, 60 is expressedabove as the sum of two powers of 2: +64 and −4. Multiplication requirestwo shift and add operations. If 60 were expressed in positive numbersof powers of 2, result would be the sum of four powers of 2 (32+16+8+4).Using all positive numbers results in 4 shift and add operations. Usingboth positive and negative numbers saves 2 shift and add operations.

For FIG. 6, the quantization levels and power of 2 factors are:

1155=1024+128+2+1

273=256+16+1

297=256+32+8+1

603=512+128+64+32−4−1

693=512+128+64−8−2−1

615=512+128−32+8−1

417=256+128+32+1

147=128+16+2+1

These levels may be adjusted manually to reduce the number of shift andadd operations by dropping the ±1 terms as follows:

1154=1024+128+2

272=256+16

296=256+32+8

604=512+128+64+32−4

694=512+128+64−8−2

616=512+128−32+8

416=256+128+32

146=128+16+2

In such manner, one shift and add operation for each quantization levelis eliminated. For example, to multiply by 1153 takes 4 shift and addoperations, but to multiply by 1154 takes only 3 shift and addoperations, saving one shift and add operation.

For FIG. 7, the quantization levels and power of 2 factors are:

672=512+128+32

48=32+16

480=512−32

528=512+16

288=256+32

 0=0

The above quantization numbers (quantization levels are shown in FIG.7), representing the pre-computed DD-CMA error terms are rapidlyretrieved from lookup table memory during the receiver operation. Due tothe efficient factoring of each quantization level into a minimal set ofpowers of 2, the retrieved DD-CMA error terms provide for efficientmultiply operations using shifts and adders.

For the purpose of improving stability by reducing jitter, it isadvantageous to set the quantization levels at the roots of the CMAerror function (the zero crossings at ±g) to zero. In such case, thequantization levels in FIG. 7, and power of 2 factors are:

672=512+128+32

0

480=512−32

528=512+16

288=256+32

0=0

FIG. 7 indicates that ±48 levels at the ±g roots of the CMA errorfunction are set to zero. In addition to improved stability, setting thequantization level to zero at the roots of the DD-CMA error function hasthe further advantage of eliminating two shift and add operations whencomputing the updated equalizer filter parameters.

What is claimed is:
 1. In a digital communications receiver including anequalization filter having a plurality of adjustable parameters, and anerror term calculator for updating said plurality of adjustableparameters, said error term calculator responsive to a received digitalsignal and the error in said received digital signal, said error termcalculator including an error function which is a function of saidreceived digital signal, a quantization method comprising: selecting aplurality of quantization levels corresponding to said received digitalsignal; computing said error function for each of said plurality ofquantization levels to form a quantized error function; and storing saidquantized error function in a lookup table.
 2. A method in accordancewith claim 1, further comprising: receiving said digital signal;retrieving a value of said quantized error function from said lookuptable responsive to said received digital signal; and using saidretrieved value of said quantized error function to update saidplurality of adjustable parameters of said equalization filter.
 3. Aquantization method in accordance with claim 2, wherein said digitalcommunications receiver further includes QAM passband signaling havingan in-phase signal component I, a quadrature phase signal component Q,an in-phase passband signal component I_(pb) and a quadrature phasepassband signal component I_(pb), wherein said step of retrieving avalue of said quantized error function from said lookup table responsiveto said received digital signal further includes the step ofapproximating I{circumflex over ( )}² ₊Q{circumflex over ( )}² by theterm I_(pb){circumflex over ( )}² ₊Q_(pb){circumflex over ( )}².
 4. Aquantization method in accordance with claim 1, wherein a quantizationlevel of said error function is selected to be a power of
 2. 5. Aquantization method in accordance with claim 1, wherein a quantizationlevel of said error function is selected to be the sum of at least twonumbers, each of which numbers is a power of
 2. 6. A quantization methodin accordance with claim 1, wherein a quantization level of saidquantized error function is selected to be the sum of at least twonumbers, wherein one of which two numbers is positive and the other ofwhich two numbers is negative and both of which two numbers have anabsolute value equal to a power of
 2. 7. A quantization method inaccordance with claim 1, wherein said error term calculator for updatingsaid plurality of adjustable parameters uses the Constant ModulusAlgorithm (CMA) error function.
 8. In a digital communications receiverincluding an equalization filter having a plurality of adjustableparameters, and an error term calculator for updating said plurality ofadjustable parameters, said error term calculator responsive to areceived digital signal and the error in said received digital signal,said error term calculator including an error function which is afunction of said received digital signal, an apparatus comprising: adigital signal receiver having an output corresponding to said receiveddigital signal; a lookup table having a respective input and output,said input of said lookup table responsive to said received digitalsignal, said output from said lookup table corresponding to said errorfunction, said lookup table containing values of said error functioncomputed by selecting a plurality of quantization levels correspondingto said received digital signal and computing said error function foreach of said plurality of quantization levels to form a quantized errorfunction, said lookup table storing said quantized error function; andthe output of said lookup table being coupled to said equalizationfilter to update said plurality of adjustable parameters of saidequalization filter using said values of said quantized error functionstored in said lookup table.
 9. A quantization apparatus in accordancewith claim 8, wherein a quantization level of said error function isselected to be a power of
 2. 10. A quantization apparatus in accordancewith claim 8, wherein a quantization level of said error function isselected to be the sum of at least two numbers, each of which numbers isa power of
 2. 11. A quantization apparatus in accordance with claim 8,wherein a quantization level of said quantized error function isselected to be the sum of at least two numbers, wherein one of which twonumbers is positive and the other of which two numbers is negative andboth of which two numbers have an absolute value equal to a power of 2.12. A quantization apparatus in accordance with claim 8, wherein saiderror term calculator for updating said plurality of adjustableparameters uses the Constant Modulus Algorithm (CMA) error function. 13.A quantization apparatus in accordance with claim 8, wherein saiddigital communications receiver further includes QAM passband signalinghaving an in-phase signal component I, a quadrature phase signalcomponent Q, an in-phase passband signal component I_(pb) and aquadrature phase passband signal component I_(pb), wherein said lookuptable is responsive to inputs I_(pb){circumflex over ( )} andQ_(pb){circumflex over ( )}, wherein the sum of I{circumflex over ( )}²₊Q{circumflex over ( )}² is approximated by the sum of I_(pb){circumflexover ( )}² ₊Q_(pb){circumflex over ( )}².